GEOMETRY UNIT 2 WORKBOOK CHAPTER 6 Quadrilaterals

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Geometry Section 6.1 Notes: Angles of Polygons A diagonal of a polygon is a segment that connects any two _____________ vertices. The vertices of polygon PQRST that are not consecutive with vertex P are vertices R and S. Therefore, polygon PQRST has two diagonals from vertex P,

PR and PS . Notice that the diagonals from vertex P separate the polygon into three triangles. The ____ of the angle measures of a polygon is the ____ of the angle measures of the triangles formed by drawing all the possible diagonals from one vertex. Since the sum of the angle measures of a triangle is 180°, we can make a table and look for a pattern to find the sum of the angle measures for any convex polygon.

# of Sides

# of diagonals from a single vertex

# of ∆s drawn (from a single vertex)

Sum of measures of interior angles

Triangle Quadrilateral Pentagon Hexagon N-gon This leads to the following theorem:

You can use the Polygon Interior Angles Sum Theorem to find the sum of the interior angles of a polygon and to find missing measures in polygons. Example 1: a) Find the sum of the measures of the interior angles of a convex nonagon.

b) Find the measure of each interior angle of parallelogram RSTU.

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Recall from Lesson 1.6 that in a regular polygon, all of the interior angles are congruent. You can use this fact and the Polygon Interior Angle Sum Theorem to find the interior angle measure of any regular polygon.

Example 2: A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the measure of one of the interior angles of the pentagon.

Given the interior angle measure of a regular polygon, you can also use the Polygon Interior Angles Sum Theorem to find a polygon’s number of sides.

Example 3: a) The measure of an interior angle of a regular polygon is 150°. Find the number of sides in the polygon.

b) The measure of an interior angle of a regular polygon is 144°. Find the number of sides in the polygon.

Does a relationship exist between the number of sides a convex polygon and the sum of its exterior angle measures? Examine the polygons below in which an exterior angle has been measured at each vertex.

Did you notice that the sum of the exterior angle measures in each case is 360°? This suggests the following theorem:

Example 4: a) Find the value of x in the diagram.

b) Find the measure of each exterior angle of a regular decagon.

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Geometry Section 6.1 Worksheet

Name: _____________________________________

For numbers 1 – 3, find the sum of the measures of the interior angles of each convex polygon. 1. 11-gon

2. 14-gon

3. 17-gon

For numbers 4 – 6, the measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon. 4. 144

5. 156

6. 160

For numbers 7 & 8, find the measure of each interior angle. 7.

8.

For numbers 9 – 14, find the measures of an exterior angle and an interior angle given the number of sides of each regular polygon. Round to the nearest tenth, if necessary. 9. 16

10. 24

11. 30

12. 14

13. 22

14. 40

15. Crystals are classified according to seven crystal systems. The basis of the classification is the shapes of the faces of the crystal. Turquoise belongs to the triclinic system. Each of the six faces of turquoise is in the shape of a parallelogram. Find the sum of the measures of the interior angles of one such face.

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16. In the Uffizi gallery in Florence, Italy, there is a room built by Buontalenti called the Tribune (La Tribuna in Italian). This room is shaped like a regular octagon. What angle do consecutive walls of the Tribune make with each other?

17. Jasmine is designing boxes she will use to ship her jewelry. She wants to shape the box like a regular polygon. In order for the boxes to pack tightly, she decides to use a regular polygon that has the property that the measure of its interior angles is half the measure of its exterior angles. What regular polygon should she use?

18. A theater floor plan is shown in the figure. The upper five sides are part of a regular dodecagon. Find m∠1.

19. Archeologists unearthed parts of two adjacent walls of an ancient castle. Before it was unearthed, they knew from ancient texts that the castle was shaped like a regular polygon, but nobody knew how many sides it had. Some said 6, others 8, and some even said 100. From the information in the figure, how many sides did the castle really have?

20. In Ms. Rickets’ math class, students made a “polygon path” that consists of regular polygons of 3, 4, 5, and 6 sides joined together as shown. a) Find m∠2 and m∠5.

b) Find m∠3 and m∠4.

c) What is m∠1?

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Geometry Section 6.2 Notes: Parallelograms I. Definition: A parallelogram is a quadrilateral ______________________________________ ____________________________________________________________________________________. Write a proof using parallelogram PQRS with the auxiliary line segment drawn.

Q

R

Given: PQRS is a parallelogram. Prove: PQ ≅ RS

P

II.

S

Parallelogram Properties: Both pairs of ____________ sides are ________________. Both pairs of ______________ sides are _________________.

Both pairs of ______________ angles are _______________. __________________ angles are __________________.

The _____________ bisect _________ _____________.

_____ pair of opposite sides are both _____________ AND ______________.

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III.

Practice:

1) For what values of x and y will each of the following be parallelogram? Be sure to explain why. a) b)

B

A

20° 18

E

2x y°

D

c)

d)

3) Find the measure in parallelogram HIJK. Explain your reasoning. a) HI = _______

c) KI = _______

e)

m JIH

C

= _______

H

b) GH = _______

d)

f)

m KIH

m HKI

84° G

= _______

K

28°

I

7 8

= _______

10

16 J 4) The diagonals of LMNO intersect at point P. What are the coordinates of P?

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Geometry Section 6.2 Worksheet

Name: _____________________________________

For numbers 1 – 4, find the value of each variable. 1.

2.

3.

4.

For numbers 5 – 8, use parallelogram RSTU to find the measure or value. 5. m∠RST

6. m∠STU

7. m∠TUR

8. b

For numbers 9 & 10, find the coordinates of the intersection of the diagonals of parallelogram PRYZ with the given vertices. 9. P(2, 5), R(3, 3), Y(–2, –3), Z(–3, –1)

10. P(2, 3), R(1, –2), Y(–5, –7), Z(–4, –2)

11. Write a paragraph proof of the following. Given: parallelogram PRST and parallelogram PQVU Prove: ∠V ≅ ∠S

12. Mr. Rodriquez used the parallelogram at the right to design a herringbone pattern for a paving stone. He will use the paving stone for a sidewalk. If m∠1 is 130°, find m∠2, m∠3, and m∠4.

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13. A walkway is made by adjoining four parallelograms as shown. Are the end segments a and e parallel to each other? Explain.

14. Four friends live at the four corners of a block shaped like a parallelogram. Gracie lives 3 miles away from Kenny. How far apart do Teresa and Travis live from each other?

15. Four soccer players are located at the corners of a parallelogram. Two of the players in opposite corners are the goalies. In order for goalie A to be able to see the three others, she must be able to see a certain angle x in her field of vision. What angle does the other goalie have to be able to see in order to keep an eye on the other three players?

16. Make a Venn diagram showing the relationship between squares, rectangles, and parallelograms.

17. On vacation, Tony’s family took a helicopter tour of the city. The pilot said the newest building in the city was the building with this top view. He told Tony that the exterior angle by the front entrance is 72°. Tony wanted to know more about the building, so he drew this diagram and used his geometry skills to learn a few more things. The front entrance is next to vertex B. a) What are the measures of the four angles of the parallelogram?

b) How many pairs of congruent triangles are there in the figure? What are they?

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Geometry Section 6.3 Notes: Tests for Parallelograms Before, we started with a parallelogram and then described its properties. Now, we will start with the properties. Fill in the blanks with what we discussed yesterday. Test yourself…try without using your notes. ___________ pairs of ____________________ sides are parallel. Both pairs of ____________________ sides are ______________. ___________ pairs of ____________________ angles are ____________. ____________________ angles are ____________________________. _________________ bisect _________ ____________. _____ ______________ of opposite sides are _____ AND _____. For 1-6, determine whether there is enough information to conclude that each quadrilateral below is a parallelogram. If not, explain why not. If so, state the reason. (Hint: Literally write down what the diagram is telling you and see if it fits a

property) 1)

3)

2)

4)

12

5)

6)

120°

12

60°

60°

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For 7 – 8, determine the value of x and y that will make the polygon a parallelogram. x+2

7) 3x

8)

70°

(3x+5)o

6

2x°

(x+3y)o

y-1

We can use the Distance, Slope, and Midpoint Formulas to determine whether a quadrilateral in a coordinate plane is a parallelogram. 9) Quadrilateral ABCD has vertices A(-3, 3), B(2, 5), C(5, 2), and D(0, 0). Determine whether the quadrilateral is a parallelogram.

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10) Quadrilateral QRST has vertices Q(–1, 3), R(3, 1), S(2, –3), and T(–2, –1). Determine whether the quadrilateral is a parallelogram.

11) Three of the vertices of a quadrilateral are given. Find the coordinates of point D so that it is a parallelogram: A(-2, -3), B(4, -3), C(3, 2)

y

x

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Geometry Section 6.3 Worksheet

Name: _____________________________________

For numbers 1 – 4, determine whether each quadrilateral is a parallelogram. Justify your answer. 1.

2.

3.

4

For numbers 5 & 6, graph each quadrilateral with the given vertices. Determine whether the figure is a parallelogram. Justify your answer with the method indicated. 5. P(–5, 1), S(–2, 2), F(–1, –3), T(2, –2); Slope Formula

6. R(–2, 5), O(1, 3), M(–3, –4), Y(–6, –2); Distance and Slope Formulas

For numbers 7 – 10, solve for x and y so that the quadrilateral is a parallelogram. 7. 8.

9.

10.

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11. The pattern shown in the figure is to consist of congruent parallelograms. How can the designer be certain that the shapes are parallelograms?

12. Nikia, Madison, Angela, and Shelby are balancing themselves on an “X”-shaped floating object. To balance themselves, they want to make themselves the vertices of a parallelogram. In order to achieve this, do all four of them have to be the same distance from the center of the object? Explain.

13. Two compass needles placed side by side on a table are both 2 inches long and point due north. Do they form the sides of a parallelogram?

14. Four jets are flying in formation. Three of the jets are shown in the graph. If the four jets are located at the vertices of a parallelogram, what are the three possible locations of the missing jet?

15. When a coordinate plane is placed over the Harrisville town map, the four street lamps in the center are located as shown. Do the four lamps form the vertices of a parallelogram? Explain.

16. Aaron is making a wooden picture frame in the shape of a parallelogram. He has two pieces of wood that are 3 feet long and two that are 4 feet long. a) If he connects the pieces of wood at their ends to each other, in what order must he connect them to make a parallelogram?

b) How many different parallelograms could he make with these four lengths of wood?

c) Explain something Aaron might do to specify precisely the shape of the parallelogram. 85

Geometry Sections 6.4 & 6.5 Notes: Rectangles, Rhombi, and Squares Practice: For 1-6, match the properties with the appropriate quadrilateral(s). ____________ 1) The diagonals are congruent ____________ 2) Both pairs of opposite sides are congruent

A) B) C) D)

Parallelogram Rectangle Rhombus Square

____________ 3) Both pairs of opposite sides are parallel ____________ 4) All angles are congruent ____________ 5) All sides are congruent ____________ 6) Diagonals bisect the opposite angles

For 7-13, fill in the blanks with always, sometimes, or never. 7. A Rhombus is ________________ equilateral. 8. The diagonals of a rectangle are ________________ perpendicular. 9. A rectangle is ________________ a rhombus. 10) A parallelogram is ________________ a rhombus. 11) A square is ________________ a rhombus. 12) A rectangle is ________________ a parallelogram. 13) A square is a ________________ rectangle.

For 14-16, answer the question and state the property that allows you to set up the equation. 14) WXYZ is a rectangle. Find the value of x. W

X

14) x = _____________ Property: ___________________ ___________________________

7x + 4

3(x + 8)

Z

Y

15) The diagonals of rhombus WXYZ intersect at V. If m∠WZX = 39.5°, find m∠ZYX.

15) m∠ZYX. = ____________ Property: ____________________ ____________________________

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16) WXYZ is a square.

WX = 1 − 10x YZ= 14 + 3x XY = ___

16) XY = _____________ Property: ___________________ ___________________________

You can also use the properties of rectangles to prove that a quadrilateral positioned on a coordinate plane is a rectangle, rhombus, or square given the coordinates of the vertices. Example 1: Quadrilateral JKLM has vertices J(–2, 3), K(1, 4), L(3, –2), and M(0, –3). Determine whether JKLM is a rectangle using the Distance Formula.

Example 2: Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for the given vertices: A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2). List all that apply. Explain.

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Geometry Section 6.4 Worksheet

Name: _____________________________________

For numbers 1 – 6, quadrilateral RSTU is a rectangle. 1. If UZ = x + 21 and ZS = 3x – 15, find US.

2. If RZ = 3x + 8 and ZS = 6x – 28, find UZ.

3. If RT = 5x + 8 and RZ = 4x + 1, find ZT.

4. If m∠SUT = (3x + 6)° and m∠RUS = (5x – 4)°, find m∠SUT.

5. If m∠SRT = (x + 9)° and m∠UTR = (2x – 44)°, find m∠UTR.

6. If m∠RSU = (x + 41)° and m∠TUS = (3x + 9)°, find m∠RSU.

For numbers 7 – 12, quadrilateral GHJK is a rectangle. Find each measure if m∠1 = 37°. 7. m∠2

8. m∠3

9. m∠4

10. m∠5

11. m∠6

12. m∠7

For numbers 13 – 15, graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula. 13. B(–4, 3), G(–2, 4), H(1, –2), L(–1, –3); Slope Formula

14. N(–4, 5), O(6, 0), P(3, –6), Q(–7, –1); Distance Formula

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15. C(0, 5), D(4, 7), E(5, 4), F(1, 2); Slope Formula

16. Huntington Park officials approved a rectangular plot of land for a Japanese Zen garden. Is it sufficient to know that opposite sides of the garden plot are congruent and parallel to determine that the garden plot is rectangular? Explain.

17. Jalen makes the rectangular frame shown. In order to make sure that it is a rectangle, Jalen measures the distances BD and AC. How should these two distances compare if the frame is a rectangle?

18. A bookshelf consists of two vertical planks with five horizontal shelves. Are each of the four sections for books rectangles? Explain.

19. A landscaper is marking off the corners of a rectangular plot of land. Three of the corners are in place as shown. What are the coordinates of the fourth corner?

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20. Antonio is designing a swimming pool on a coordinate grid. Is it a rectangle? Explain.

21. Veronica made the pattern shown out of 7 rectangles with four equal sides. The side length of each rectangle is written inside the rectangle. a) How many rectangles can be formed using the lines in this figure?

b) If Veronica wanted to extend her pattern by adding another rectangle with 4 equal sides to make a larger rectangle, what are the possible side lengths of rectangles that she can add?

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Geometry Section 6.5 Worksheet

Name: _____________________________________

For numbers 1 – 4, PRYZ is a rhombus. If RK = 5, RY = 13 and m∠YRZ = 67°, find each measure. 1. KY 2. PK 3. m∠YKZ 4. m∠PZR

For numbers 5 – 8, MNPQ is a rhombus. If PQ = 3√2 and AP = 3, find each measure. 5. AQ 6. m∠APQ 7. m∠MNP 8. PM

For numbers 9 – 11, use the given set of vertices to determine whether apply. Explain. 9. B(–9, 1), E(2, 3), F(12, –2), G(1, –4)

BEFG is a rhombus, a rectangle, or a square. List all that

10. B(1, 3), E(7, –3), F(1, –9), G(–5, –3)

11. B(–4, –5), E(1, –5), F(–2, –1), G(–7, –1)

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12. The figure is an example of a tessellation. Use a ruler or protractor to measure the shapes and then name the quadrilaterals used to form the figure.

13. A tray rack looks like a parallelogram from the side. The levels for the trays are evenly spaced. What two labeled points form a rhombus with base AA' ?

14. Charles cuts a rhombus along both diagonals. He ends up with four congruent triangles. Classify these triangles as acute, obtuse, or right.

15. The edges of a window are drawn in the coordinate plane. Determine whether the window is a square or a rhombus.

16. Mackenzie cut a square along its diagonals to get four congruent right triangles. She then joined two of them along their long sides. Show that the resulting shape is a square.

17. Tatianna made the design shown. She used 32 congruent rhombi to create the flower-like design at each corner. a) What are the angles of the corner rhombi?

b) What kinds of quadrilaterals are the dotted and checkered figures?

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Geometry 6.6 Notes: Trapezoids and Kites A trapezoid is a quadrilateral with exactly ____ pair of parallel sides. The __________ sides are called bases. The nonparallel sides are called legs. The base angles are formed by the ______ and one of the legs. In trapezoid ABCD, ∠A and ∠B are one pair of base angles and ∠C and ∠D are the other pair. If the legs of a trapezoid are congruent, then it is an isosceles trapezoid.

PRACTICE IT!! For #1-5, match the segments/angles with the term which best describes them in trapezoid PQRS. ______ 1) ______ 2) ______ 3) ______ 4) ______ 5)

QR and PS PQ and RS QS and PR

Q and S

S and P

Q

A. bases

R

B. legs C. diagonals D. base angles E. opposite angles

COIN IT!!!

P

S PICTURE IT!!

A trapezoid is a quadrilateral with…

The midsegment of a trapezoid is a segment that…

If the legs of a trapezoid are congruent, then the trapezoid is…

THEOREM IT!! If a trapezoid is isosceles, then each pair of base angles is congruent. If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid. A trapezoid is isosceles if and only if its diagonals are congruent.

The midsegment of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases.

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Example 1: Quadrilateral ABCD has vertices A(5, 1), B(–3, –1), C(–2, 3), and D(2, 4). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid.

Example 2: Each side of the basket shown is an isosceles trapezoid. If KN = 6.7 feet, and LN = 3.6 feet, find MK.

Example 3: Find the measure of all angles in ABCD for each trapezoid below. a)

A

b)

B

53°

Example 4: The midsegment of the trapezoid is a)

91°

132°

c)

B

RT

C

c)

7 x

D

. Find the value of x.

b)

R

B

A 108°

C

D

C

D

A

R

8

T

R

11

T 24

14

13

x

x

T

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Flyin’ with Kites!!

Notes! If you place two non-congruent isosceles ∆s together (base-to-base), you can make an exciting new quadrilateral called a __________________________________!!!!!!

B

A 1) In the kite to the right, which sides appear to be congruent?

2) Are the congruent sides opposite or consecutive?

C

D

3) Draw in the altitudes of the ∆s and include the appropriate  marks. What must be true about the diagonals of the kite? 4) Which  s of the ∆ are ≅ ? 5) Which  s of the kite are ≅ ? Definition of a Kite:

Theorem:

Theorem: #2 Ex. A: Circle which of the diagrams on this page are geometrical kites!! 96

Examples: For each kite below, find the length of the sides or the measure of each angle.

a)

b)

c)

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Geometry Section 6.6 Worksheet

Name: _____________________________________

For numbers 1 – 4, find each measure. 1. m∠T

3. m∠Q

2. m∠Y

4. BC

For numbers 5 & 6, use trapezoid FEDC, where V and Y are midpoints of the legs. 5. If FE = 18 and VY = 28, find CD.

6. If m∠F = 140° and m∠E = 125°, find m∠D.

For numbers 7 & 8, RSTU is a quadrilateral with vertices R(–3, –3), S(5, 1), T(10, –2), U(–4, –9). 7. Verify that RSTU is a trapezoid.

8. Determine whether RSTU is an isosceles trapezoid. Explain.

9. A set of stairs leading to the entrance of a building is designed in the shape of an isosceles trapezoid with the longer base at the bottom of the stairs and the shorter base at the top. If the bottom of the stairs is 21 feet wide and the top is 14 feet wide, find the width of the stairs halfway to the top.

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10. A carpenter needs to replace several trapezoid-shaped desktops in a classroom. The carpenter knows the lengths of both bases of the desktop. What other measurements, if any, does the carpenter need?

11. Artists use different techniques to make things appear to be 3-dimensional when drawing in two dimensions. Kevin drew the walls of a room. In real life, all of the walls are rectangles. In what shape did he draw the side walls to make them appear 3-dimensional?

12. In order to give the feeling of spaciousness, an architect decides to make a plaza in the shape of a kite. Three of the four corners of the plaza are shown on the coordinate plane. If the fourth corner is in the first quadrant, what are its coordinates?

13. A simplified drawing of the reef runway complex at Honolulu International Airport is shown below. How many trapezoids are there in this image?

14. A light outside a room shines through the door and illuminates a trapezoidal region ABCD on the floor. Under what circumstances would trapezoid ABCD be isosceles?

15. A riser is designed to elevate a speaker. The riser consists of 4 trapezoidal sections that can be stacked one on top of the other to produce trapezoids of varying heights. All of the stages have the same height. If all four stages are used, the width of the top of the riser is10 feet. a) If only the bottom two stages are used, what is the width of the top of the resulting riser?

b) What would be the width of the riser if the bottom three stages are used?

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